On Finsler metric measure manifolds with integral weighted Ricci curvature bounds

TL;DR

This paper investigates the geometric and topological properties of Finsler metric measure manifolds under integral weighted Ricci curvature bounds, establishing key comparison theorems, volume estimates, and eigenvalue bounds.

Contribution

It introduces new comparison theorems and estimates for Finsler manifolds with integral weighted Ricci curvature bounds, expanding understanding beyond pointwise bounds.

Findings

Laplacian comparison theorem established

Volume comparison theorems proved

Eigenvalue and gradient estimates derived

Abstract

In this paper, we study deeply geometric and topological properties of Finsler metric measure manifolds with the integral weighted Ricci curvature bounds. We first establish Laplacian comparison theorem, Bishop-Gromov type volume comparison theorem and relative volume comparison theorem on such Finsler manifolds. Then we obtain a volume growth estimate and Gromov pre-compactness under the integral weighted Ricci curvature bounds. Furthermore, we prove the local Dirichlet isoperimetric constant estimate on Finsler metric measure manifolds with integral weighted Ricci curvature bounds. As applications of the Dirichlet isoperimetric constant estimates, we get first Dirichlet eigenvalue estimate and a gradient estimate for harmonic functions.